If `(x+1/x)=3` , find the value of `(x^3-1/x^3)(x^4 + 1/x^4)`

Options

A. 846
B. `235sqrt5`
C. `376sqrt5`
D. `329sqrt5`

Answer

To find the value of \((x^3 - \frac{1}{x^3})(x^4 + \frac{1}{x^4})\) given that \(x + \frac{1}{x} = 3\), we can follow these steps:

### Step 1: Find \(x^2 + \frac{1}{x^2}\)
We start with the equation:
\[ x + \frac{1}{x} = 3 \]
Now, we square both sides:
\[ \left(x + \frac{1}{x}\right)^2 = 3^2 \]
This expands to:
\[ x^2 + 2 + \frac{1}{x^2} = 9 \]
Subtracting 2 from both sides gives:
\[ x^2 + \frac{1}{x^2} = 7 \]

### Step 2: Find \(x^3 - \frac{1}{x^3}\)
We can use the identity:
\[ x^3 - \frac{1}{x^3} = \left(x + \frac{1}{x}\right)\left(x^2 - 1 + \frac{1}{x^2}\right) \]
We already know \(x + \frac{1}{x} = 3\) and we can find \(x^2 - 1 + \frac{1}{x^2}\) as follows:
\[ x^2 - 1 + \frac{1}{x^2} = (x^2 + \frac{1}{x^2}) - 1 = 7 - 1 = 6 \]
Now substituting back:
\[ x^3 - \frac{1}{x^3} = 3 \cdot 6 = 18 \]

### Step 3: Find \(x^4 + \frac{1}{x^4}\)
We can use the identity:
\[ x^4 + \frac{1}{x^4} = \left(x^2 + \frac{1}{x^2}\right)^2 - 2 \]
Substituting \(x^2 + \frac{1}{x^2} = 7\):
\[ x^4 + \frac{1}{x^4} = 7^2 - 2 = 49 - 2 = 47 \]

### Step 4: Calculate \((x^3 - \frac{1}{x^3})(x^4 + \frac{1}{x^4})\)
Now we can find the final result:
\[ (x^3 - \frac{1}{x^3})(x^4 + \frac{1}{x^4}) = 18 \cdot 47 \]
Calculating this gives:
\[ 18 \cdot 47 = 846 \]

### Final Answer
Thus, the value of \((x^3 - \frac{1}{x^3})(x^4 + \frac{1}{x^4})\) is:
\[ \boxed{846} \]